I (of course) have the same issue as the other posters - the dreaded 3689 message. We have two Apple TVs and two Windows computers (previously XP, now 7) that have worked reasonably well for the past couple of years, although I do have to say I've found the Apple TVs to be one of the buggiest hardware products I've had the misfortune to work with - especially grating since Apple is supposed to be "it just works". I'm not an IT guru, but I did install and maintain our office network with 10 computers under Windows Server, so I know my way around routers/firewalls/ports, etc.
Anyway, maybe my info will be of help to others. Both Apple TVs worked fine up until about three days ago. Now both Apple TVs give the 3689 message (started at the same time). This of course implies the problem is with Itunes or the router. There have been NO changes in the router and NO changes to Itunes (aside from updating to 8.2.1). I tried syncing both ATVS to the other computer and got the same message. Having a couple years under my belt with the Apple TVs, I have:
1) double-checked router firewall settings and disabled windows firewall. I even turned off my router firewall for a few minutes so the network was completely open, and still got the Port 3689 message - I have NO antivirus or other potential port-blocking programs running. I'm quite sure this is just a catch-all "hey I can't talk to the Apple TV" error and the Port 3689 message is misleading since it implies that's the only possible problem.
2) rebooted computer many times;
3) restored ATVs to factory settings and reinstalled all updates;
4) rebooted router many times;
5) disconnected and reconnected both ATVs many times;
6) completely cleared out Itunes and the library and reinstalled them from scratch - deleted the XML file containing library info and started from scratch.
The only other things I can think of are:
1) We both got Iphones and maybe trying to sync the Iphones to Itunes permanently confused Itunes, but I turned off both Iphones and disconnected them before reinstalling Itunes, so I don't think that's it.
2) I switched the HDMI cables on the two ATVs - dunno how this could cause anything, but a previous poster mentioned HDMI so thought I'd toss it out.
3) updated Itunes and ATVs to the latest update (most likely problem).
If I manage to find out what the issue is, I'll post back. But be aware that the 3689 message can mean many different problems, which is why people are saying how multiple different solutions worked for them. Don't automatically assume it's a port issue, although it's one possibility - could be almost anything, unfortunately.